From five boys named A, B, C, D, and E, groups of three are to be formed such that no group includes both C and D simultaneously. What is the greatest number of such distinct groups possible?

Explanation

The total number of ways to choose 3 boys out of 5 is 10 (5 choose 3). However, groups containing both C and D are not allowed. The groups that include both C and D are: C, D, A; C, D, B; and C, D, E, totaling 3 groups. Subtracting these from 10 gives 7 valid groups. Hence, the maximum number of groups without both C and D together is 7.